Abstract
Microscale fluid dynamics has received intensive interest due to the emergence of Micro-Electro-Mechanical Systems (MEMS) technology. When the mean free path of the gas is comparable to the channel’s characteristic dimension, the continuum assumption is no longer valid and a velocity slip may occur at the duct walls. Non-circular cross sections are common channel shapes that can be produced by microfabrication. The non-circular microchannels have extensive practical applications in MEMS. Slip flow in non-circular microchannels has been examined and a simple model is proposed to predict the friction factor and Reynolds product fRe for slip flow in most non-circular microchannels. Through the selection of a characteristic length scale, the square root of cross-sectional area, the effect of duct shape has been minimized. The developed model has an accuracy of 10% for most common duct shapes. The developed model may be used to predict mass flow rate and pressure distribution of slip flow in non-circular microchannels.
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Abbreviations
- A :
-
flow area (m2)
- a :
-
major semi-axis of ellipse or rectangle (m)
- a :
-
base width of a trapezoidal or double-trapezoidal duct (m)
- b :
-
minor semi-axis of ellipse or rectangle (m)
- b :
-
height of a trapezoidal or double-trapezoidal duct (m)
- c :
-
half focal length of ellipse (m)
- c :
-
short side of a trapezoidal or double-trapezoidal duct (m)
- D h :
-
hydraulic diameter = 4A/P
- E(e):
-
complete elliptical integral of the second kind
- e :
-
eccentricity \(={{\sqrt{1 - {b^{2}} \mathord{\left/ {\vphantom {{b^{2}} {a^{2}}}} \right. \kern-\nulldelimiterspace} {a^{2}}}}}\)
- f :
-
Fanning friction factor \(={\tau /{\left({\tfrac{1}{2}\rho \bar{u}^{2}} \right)}}\)
- Kn :
-
Knudsen number \({=\lambda \mathord{\left/{\vphantom {\lambda {\ell}}} \right. \kern-\nulldelimiterspace} {\ell}}\)
- Kn * :
-
modified Knudsen number =Kn(2−σ)/σ
- L :
-
channel length (m)
- L + :
-
dimensionless channel length \({=L \mathord{\left/ {\vphantom {L {D_{h} {Re}}}} \right. \kern-\nulldelimiterspace} {D_{\rm h} {Re}}_{{D_{\rm h}}}}\)
- ℓ:
-
arbitrary length scale
- Ma :
-
Mach number = u/V s
- \({\dot{m}}\) :
-
mass flow rate (kg/s)
- P :
-
perimeter (m)
- Po :
-
Poiseuille number, \({={\bar{\tau}{\kern 1pt} {\ell}} \mathord{\left/ {\vphantom {{\bar{\tau}{\kern 1pt} {\ell}} {\mu {\kern 1pt} \bar{u}}}} \right. \kern-\nulldelimiterspace} {\mu {\kern 1pt} \bar{u}}}\)
- p :
-
pressure \({N \mathord{\left/{\vphantom {{\rm N} {{\rm m}^{2}}}} \right. \kern-\nulldelimiterspace} {m^{2}}}\)
- R :
-
specific gas constant \({J \mathord{\left/ {\vphantom {{\rm J} {{\rm kg}{\kern 1pt} K}}} \right. \kern-\nulldelimiterspace} {kg{\kern 1pt} K}}\)
- Re :
-
Reynolds number = \({{\ell}\bar{u}/\nu}\)
- r :
-
dimensionless radius ratio = r i /r o
- r i :
-
inner radius of a concentric duct (m)
- r o :
-
outer radius of a concentric duct (m)
- T :
-
temperature (K)
- U :
-
velocity scale (m/s)
- u :
-
velocity (m/s)
- \({\bar{u}}\) :
-
average velocity (m/s)
- V s :
-
speed of sound \({={\sqrt{\gamma RT}}}\)
- X n :
-
function of x/a
- x, y :
-
Cartesian coordinates (m)
- z :
-
coordinate in flow direction (m)
- α:
-
constants
- γ:
-
ratio of specific heats
- δ n :
-
eigenvalues
- ε:
-
aspect ratio =b/a
- η, ψ, z :
-
elliptic cylinder coordinates
- η0 :
-
parameter of elliptic cylinder coordinates
- λ:
-
molecular mean free path (m)
- μ:
-
dynamic viscosity \({{N{\kern 1pt} s} \mathord{\left/ {\vphantom {{{\rm N}{\kern 1pt} {\rm s}} {{\rm m}^{2}}}} \right. \kern-\nulldelimiterspace} {m^{2}}}\)
- ν:
-
kinematic viscosity (m2/s)
- σ:
-
tangential momentum accommodation coefficient
- τ:
-
wall shear stress \({{N{\kern 1pt}} \mathord{\left/ {\vphantom {{{\rm N}{\kern 1pt}} {{\rm m}^{2}}}} \right. \kern-\nulldelimiterspace} {m^{2}}}\)
- Φ:
-
half angle rad
- \({{\sqrt{A}}}\) :
-
based upon the square root of flow area
- c :
-
continuum
- D h :
-
based upon the hydraulic diameter
- i :
-
inlet
- ℓ:
-
based upon the arbitrary length ℓ
- o :
-
outlet
- E :
-
ellipse
- R :
-
rectangle
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The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Duan, Z., Muzychka, Y.S. Slip flow in non-circular microchannels. Microfluid Nanofluid 3, 473–484 (2007). https://doi.org/10.1007/s10404-006-0141-4
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DOI: https://doi.org/10.1007/s10404-006-0141-4